Controllability Properties of Numerical Eigenvalue Algorithms

Controllability Properties of Numerical Eigenvalue Algorithms Uwe Helmke1 and Fabian Wirth2 ? 1 2

Mathematisches Institut, Universitat Wurzburg, 97074 Wurzburg, Germany,

helmke@@mathematik.uni-wuerzburg.de

Zentrum fur Technomathematik, Universitat Bremen, 28334 Bremen, Germany, fabian@@math.uni-bremen.de

Abstract. We analyze controllability properties of the inverse iteration and the

QR-algorithm equipped with a shifting parameter as a control input. In the case of the inverse iteration with real shifts the theory of universally regular controls may be used to obtain necessary and sucient conditions for complete controllability in terms of the solvability of a matrix equation. Partial results on conditions for the solvability of this matrix equation are given. We discuss an interpretation of the system in terms of control systems on rational functions. Finally, rst results on the extension to inverse Rayleigh iteration on Grassmann manifolds using complex shifts is discussed.

For many numerical matrix eigenvalue methods such as the QR algorithm or inverse iterations shift strategies have been introduced in order to design algorithms that have faster (local) convergence. The shifted inverse iteration is studied in [3,4,15] and in [17,18], where the latter references concentrate on complex shifts. For an algorithm using multidimensional shifts for the QRalgorithm see the paper of Absil, Mahony, Sepulchre and van Dooren in this book. In this paper we interpret the shifts as control inputs to the algorithm. With this point of view standard shift strategies as the well known Rayleigh iteration can be interpreted as feedbacks for the control system. It is known (for instance in the case of the inverse iteration or its multidimensional analogue, the QR-algorithm) that the behavior of the Rayleigh shifted algorithm can be very complicated, in particular if it is applied to non-Hermitian matrices A [4]. It is therefore of interest to obtain a better understanding of the underlying control system, which up to now has been hardly studied. Here we focus on controllability properties of the corresponding systems on projective space for the case of inverse iteration, respectively the Grassmannian manifold for the QR-algorithm. As it turns out the results depend heavily on the question whether one uses real or complex shifts. The controllability of the inverse iteration with complex shifts has been studied in [13], while the real case is treated in [14]. ?

This paper was written while Fabian Wirth was a guest at the Centre Automatique et Systemes, Ecole des Mines de Paris, Fontainebleau, France. The hospitality of all the members of the centre is gratefully acknowledged.

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Uwe Helmke, Fabian Wirth

In Section 1 we introduce the shifted inverse power iteration with real shifts and the associated system on projective space and discuss its forward accessibility properties. In particular, there is an easy characterization of the set of universally regular control sequences, that is those sequences with the property, that they steer every point into the interior of its forward orbit. This will be used in Section 2 to give a characterization of complete controllability of the system on projective space in terms of solvability of a matrix equation. In Section 3 we investigate the obtained characterization and interpret it in terms of the characteristic polynomial of A. Some concrete cases in which it is possible to decide based on spectral information whether a matrix leads to complete controllable shifted inverse iteration are presented in Section 4. An interpretation of these results in terms of control systems on rational functions is given in Section 5. In Section 6 we turn to the analysis of the shifted QR algorithm. We show that the corresponding control system on the Grassmannian is never controllable except for few cases. The reachable sets are characterized in terms of Grassmann simplices. We conclude in Section 7.

1 The shifted inverse iteration on projective space We begin by reviewing recent results on the shifted inverse iteration which will motivate the ideas employed in the case of the shifted QR algorithm. Let A denote a real n  n-matrix with spectrum (A)  C . The shifted inverse iteration in its controlled form is given by

? ut I )?1 x(t) ; t 2 N ; x(t + 1) = k((A A ? u I )?1 x(t)k t

(1)

where ut 2= (A). This describes a nonlinear control system on the (n ? 1)sphere. The trajectory corresponding to a normalized initial condition x0 and a control sequence u = (u0 ; u1 ; : : : ) is denoted by (t; x0 ; u). Via the choice ut = x (t)Ax(t) we obtain from (1) the Rayleigh quotient iteration studied in [3], [4]. If the initial condition x0 for system (1) lies in an invariant subspace of A then the same holds true for the entire trajectory (t; x0 ; u), regardless of the control sequence u. In order to understand the controllability properties from x0 it would then suce to study the system in the corresponding invariant subspace. Therefore we may restrict our attention to those points not lying in a nontrivial invariant subspace of A, i.e. those x 2 Rn such that fx; Ax; : : : ; An?1 xg is a basis of Rn . Vectors with this property are called cyclic and a matrix A is called cyclic if it has a cyclic vector, which we will always assume in the following. To keep notation short let us introduce the union of A-invariant subspaces

V (A) :=

[

AV V;0